1. Field of the Invention
The present invention relates to a multiplexing optical system. More particularly, the present invention relates to an apparatus capable of performing a plurality of filtering operations simultaneously on an input image and obtaining a result of the filtering process.
2. Discussion of Related Art
In the field of image processing, frequency filtering is frequently executed to change the spatial frequency distribution of an input image for the purpose of emphasizing a part of the input image or extracting only a specific component which is obscured by noise. To perform such frequency filtering, a Fourier transformed image of the original image must be obtained. Let us express the amplitude distribution of an input image by f(x,y) and the Fourier transform F(.mu.,.nu.) thereof by EQU F{f(x,y)}=F(.mu.,.nu.) (1)
When the Fourier transform F(.mu.,.nu.) is subjected to filtering expressed by the function H(.mu.,.nu.), the following relationship holds: EQU F{F(.mu.,.nu.)H(.mu.,.nu.)}=h(-x, -y)f(-x,-y) (2)
where "*" represents convolution calculation.
In the above expression, H(.mu.,.nu.) is the Fourier transform of h(x,y). In other words, the convolution of the input image f(-x,-y) with h(-x,-y) is the Fourier transform of the product of their respective Fourier transforms F(.mu.,.nu.) and H(.mu.,.nu.). However, this processing requires an exceedingly large quantity of computation. Therefore, in the case of sequential processing as in an electronic computer, it takes a long time to process data having a massive amount of information such as two-dimensional images.
Meanwhile, by virtue of its high-speed nature and parallelism, light makes it possible to obtain a Fourier transformed image of a two-dimensional image, which has a large amount of information, at a high speed which is absolutely impossible to attain with an electronic computer. By disposing a filter corresponding to H(.mu.,.nu.) in a plane where the Fourier transformed image is formed, filtering in the frequency space of the image can be readily performed. Optical systems that perform such filtering are shown in FIGS. 12 and 13. It should be noted that in the following description a side where principal rays enter a lens is referred to as the "front side", and a side where the principal rays exit from the lens is referred to as the "back side".
First, the optical system shown in FIG. 12 will be explained. Light from a light source 11 passes successively through a condenser lens 12 and a collimator lens 13 to form collimated light having an enlarged beam width. The collimated light enters a lens array 14 having a focal length f.sub.a. A lens 15 has a focal length f.sub.2. The distance between the lens array 14 and the lens 15 is equal to the sum of the focal lengths f.sub.a and f.sub.2. Consequently, light beams emanating from the lens 15 form collimated light, and the light beams are incident from various directions on an input plane F1, which is the back focal plane of the lens 15. A spatial light modulator 21 is disposed such that the read surface thereof is coincident with the input plane F1. An input image f(x,y) 211 is displayed on the read surface of the spatial light modulator 21. A Fourier transform lens 31 is disposed such that the front focal plane thereof is coincident with the read surface of the spatial light modulator 21. Therefore, each light beam from the spatial light modulator 21 forms a Fourier transformed image F(.mu.,.nu.) 311 of the input image f(x,y) 211 on a Fourier transform plane F2, which is the back focal plane of the Fourier transform lens 31.
The above-described processing is carried out for each parallel light beam formed by the combination of the lens array 14 and the lens 15. Accordingly, a plurality of Fourier transformed images F(.mu.,.nu.) 311 of the input image f(x,y) 211 are reproduced on the Fourier transform plane F2. A lens array 331 for performing an inverse Fourier transform on each of the reproduced Fourier transformed images 311 is placed such that the front focal plane thereof is coincident with the Fourier transform plane F2. Consequently, the input image 211 is reproduced on a reproducing plane F3, which is the back focal plane of each lens element 91 of the lens array 331. In this optical system, a variety of different filters 321H.sub.i (.mu.,.nu.) (i=1, 2, 3 . . . ) are disposed for a plurality of Fourier transformed images 311 formed on the Fourier transform plane F2, and thus filtered reproduced images F{F(.mu.,.nu.)H.sub.i (.mu.,.nu.)}=h.sub.i *f are formed on the reproducing plane F3.
There has also been proposed an optical system such as that shown in FIG. 13 (Dec. 10, 1995/Vol.34, No.35/Applied Optics). A parallel light beam emitted from a light source 11 reads an input image f(x,y) 221 on an input plane F1 and then enters a Damman grating 23. The Damman grating 23 is a grating with a binary transmittance which is designed so that some different orders of diffracted light have a uniform intensity. The Damman grating 23 causes diffracted light to enter a Fourier transform lens 31 at an angle unique to each order of diffraction, and the number of Fourier transformed images F(.mu.,.nu.) 311 of the input image 221 which is equal to the number of orders of diffraction are reproduced for each order of diffraction on a Fourier transform plane F2, which is the back focal plane of the Fourier transform lens 31. On the Fourier transform plane F2, each of the reproduced Fourier transformed images F(.mu.,.nu.) 311 is filtered by each matched filter H.sub.i (.mu.,.nu.) of a matched filter array 322 having different Fourier transforms recorded for respective filters. Further, an inverse Fourier transform lens 332 forms filtered images of the input image on a reproducing plane F3.
If each matched filter of the matched filter array 322 is a spatial frequency cutoff filter, light beams from each matched filter form convolution images at the same positions in the reproducing plane F3 by the action of the inverse Fourier transform lens 332. However, if each matched filter of the matched filter array 322 is a spatial frequency cutoff filter, a plurality of convolution images overlap each other on the reproducing plane F3, making it impossible to obtain desired images. Therefore, to prevent this problem, a holographic filter is used as each matched filter of the matched filter array 322, and a result of the convolution of the input image 221 with the filter is obtained by performing an inverse Fourier transform on -1st-order diffracted light through the inverse Fourier transform lens 332. The direction of -1st-order diffraction of the holographic filter used as each matched filter is effectively adjusted so that a plurality of reproduced convolution images do not overlap each other on the reproducing plane F3.
However, the arrangement of the first conventional optical system, which is shown in FIG. 12, needs to prepare two lens arrays 14 and 331 for producing parallel light beams different in the read direction from each other to read the input image 211 displayed on the spatial light modulator 21 and for performing an inverse Fourier transform on each Fourier transformed image 311 formed on the Fourier transform plane F2. It costs a great deal to produce a lens array with high accuracy in terms of the pitch between the lens elements and the optical performance of the lens, and it takes a great deal of effort to effect alignment for the entire optical system. Further, the whole lens array 14 must be illuminated in order to produce each parallel light beam for reading the input image 211. Therefore, it is necessary to diverge the light beam to a considerable extent by the combination of the condenser lens 12 and the collimator lens 13. Consequently, the distance between the condenser lens 12 and the collimator lens 13 increases, and it is necessary to lengthen the optical system comprising the lens array 14 and the lens 15. As a result, the entire optical system undesirably becomes exceedingly long.
The second conventional optical system, which is shown in FIG. 13, has a simple arrangement in which a plurality of parallel light beams are obtained by using the Damman grating 23. However, because each matched filter used to form a matched filter array is a holographic filter, the amount of light for forming an image filtered and reproduced on the reproducing plane F3 depends on the diffraction efficiency of the holographic filter. Accordingly, the light intensity for the image formation is very weak in the present state of art. In addition, because zero-order diffracted light after the matched filter array is distributed about a point of intersection of the reproducing plane F3 and the optical axis z, the zero-order light distributed in the center of the reproducing plane F3 must be avoided when each reproduced image is detected. Accordingly, the image of zero-order light interferes with an image pickup operation for taking a plurality of filtered images. Thus, the processing efficiency is extremely low.
The direction of propagation of a light beam after the computation by each matched filter formed from a holographic filter is determined by the incident angle of the matched filter when produced. In this regard, it is extremely difficult to determine the incident angle precisely.